Remark. When X is a “simple” space, such as ℝ or ℂ a zero is also called a root. However, in pure mathematics and especially if Z⁢(f) is infinite, it seems to be customary to talk of zeroes and the zero set instead of roots.

Examples

•

For any z∈ℂ, define z^:X→ℂ by z^⁢(x)=z. Then Z⁢(0^)=X and Z⁢(z^)=∅ if z≠0.

•

Suppose p is a polynomial (http://planetmath.org/Polynomial) p:ℂ→ℂ of degree n≥1. Then p has at most n zeroes. That is, |Z⁢(p)|≤n.

•

If f and g are functions f:X→ℂ and g:X→ℂ, then

Z⁢(f⁢g)

=

Z⁢(f)∪Z⁢(g),

Z⁢(f⁢g)

⊇

Z⁢(f),

where f⁢g is the function x↦f⁢(x)⁢g⁢(x).

•

For any f:X→ℝ, then

Z⁢(f)=Z⁢(|f|)=Z⁢(fn),

where fn is the defined fn⁢(x)=(f⁢(x))n.

•

If f and g are both real-valued functions, then

Z⁢(f)∩Z⁢(g)=Z⁢(f2+g2)=Z⁢(|f|+|g|).

•

If X is a topological space and f:X→ℂ is a function, then the support (http://planetmath.org/SupportOfFunction) of f is given by: